Theorem dfdm3 4791

Description: Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Assertion

Ref	Expression
dfdm3	|- dom A = { x | E. y <. x , y >. e. A }

Distinct variable group:   x, y, A

Proof of Theorem dfdm3

Step	Hyp	Ref	Expression
1		df-dm 4614	. 2 |- dom A = { x | E. y x A y }
2		df-br 3983	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3	2	exbii 1593	. . 3 |- ( E. y x A y <-> E. y <. x , y >. e. A )
4	3	abbii 2282	. 2 |- { x | E. y x A y } = { x | E. y <. x , y >. e. A }
5	1, 4	eqtri 2186	1 |- dom A = { x | E. y <. x , y >. e. A }

Syntax hints:   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   <.cop 3579   class class class wbr 3982   dom cdm 4604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-br 3983  df-dm 4614

This theorem is referenced by:  csbdmg  4798